3.44 \(\int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=262 \[ -\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac{2431 a^2 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{2016 d}-\frac{221 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{640 d}+\frac{2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{2431 a^8 x}{256}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d} \]

[Out]

(2431*a^8*x)/256 - (2431*a^8*Cos[c + d*x]^3)/(384*d) + (2431*a^8*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (17*a^3*
Cos[c + d*x]^3*(a + a*Sin[c + d*x])^5)/(48*d) - (17*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^6)/(90*d) - (a*Cos
[c + d*x]^3*(a + a*Sin[c + d*x])^7)/(10*d) - (2431*a^2*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^3)/(2016*d) - (
221*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^4)/(336*d) - (2431*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x])^2)/(112
0*d) - (2431*Cos[c + d*x]^3*(a^8 + a^8*Sin[c + d*x]))/(640*d)

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Rubi [A]  time = 0.374343, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2678, 2669, 2635, 8} \[ -\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac{2431 a^2 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{2016 d}-\frac{221 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{640 d}+\frac{2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{2431 a^8 x}{256}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]

[Out]

(2431*a^8*x)/256 - (2431*a^8*Cos[c + d*x]^3)/(384*d) + (2431*a^8*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (17*a^3*
Cos[c + d*x]^3*(a + a*Sin[c + d*x])^5)/(48*d) - (17*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^6)/(90*d) - (a*Cos
[c + d*x]^3*(a + a*Sin[c + d*x])^7)/(10*d) - (2431*a^2*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^3)/(2016*d) - (
221*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^4)/(336*d) - (2431*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x])^2)/(112
0*d) - (2431*Cos[c + d*x]^3*(a^8 + a^8*Sin[c + d*x]))/(640*d)

Rule 2678

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx &=-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac{1}{10} (17 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^7 \, dx\\ &=-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac{1}{6} \left (17 a^2\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^6 \, dx\\ &=-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}+\frac{1}{48} \left (221 a^3\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^5 \, dx\\ &=-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}+\frac{1}{336} \left (2431 a^4\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx\\ &=-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}+\frac{1}{224} \left (2431 a^5\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}+\frac{1}{160} \left (2431 a^6\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac{1}{128} \left (2431 a^7\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac{1}{128} \left (2431 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}+\frac{2431 a^8 \cos (c+d x) \sin (c+d x)}{256 d}-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}+\frac{1}{256} \left (2431 a^8\right ) \int 1 \, dx\\ &=\frac{2431 a^8 x}{256}-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}+\frac{2431 a^8 \cos (c+d x) \sin (c+d x)}{256 d}-\frac{2431 a^5 \cos ^3(c+d x) (a+a \sin (c+d x))^3}{2016 d}-\frac{17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac{a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac{221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d}\\ \end{align*}

Mathematica [A]  time = 1.51426, size = 191, normalized size = 0.73 \[ -\frac{a^8 \left (1531530 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )+\sqrt{\sin (c+d x)+1} \left (8064 \sin ^{10}(c+d x)+63616 \sin ^9(c+d x)+209552 \sin ^8(c+d x)+353648 \sin ^7(c+d x)+257704 \sin ^6(c+d x)-130728 \sin ^5(c+d x)-492846 \sin ^4(c+d x)-543442 \sin ^3(c+d x)-410693 \sin ^2(c+d x)-508859 \sin (c+d x)+1193984\right )\right ) \cos ^3(c+d x)}{80640 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]

[Out]

-(a^8*Cos[c + d*x]^3*(1531530*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c +
d*x]]*(1193984 - 508859*Sin[c + d*x] - 410693*Sin[c + d*x]^2 - 543442*Sin[c + d*x]^3 - 492846*Sin[c + d*x]^4
- 130728*Sin[c + d*x]^5 + 257704*Sin[c + d*x]^6 + 353648*Sin[c + d*x]^7 + 209552*Sin[c + d*x]^8 + 63616*Sin[c
+ d*x]^9 + 8064*Sin[c + d*x]^10)))/(80640*d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))

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Maple [A]  time = 0.05, size = 480, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+a*sin(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/10*sin(d*x+c)^7*cos(d*x+c)^3-7/80*sin(d*x+c)^5*cos(d*x+c)^3-7/96*sin(d*x+c)^3*cos(d*x+c)^3-7/128*
cos(d*x+c)^3*sin(d*x+c)+7/256*cos(d*x+c)*sin(d*x+c)+7/256*d*x+7/256*c)+8*a^8*(-1/9*sin(d*x+c)^6*cos(d*x+c)^3-2
/21*sin(d*x+c)^4*cos(d*x+c)^3-8/105*sin(d*x+c)^2*cos(d*x+c)^3-16/315*cos(d*x+c)^3)+28*a^8*(-1/8*sin(d*x+c)^5*c
os(d*x+c)^3-5/48*sin(d*x+c)^3*cos(d*x+c)^3-5/64*cos(d*x+c)^3*sin(d*x+c)+5/128*cos(d*x+c)*sin(d*x+c)+5/128*d*x+
5/128*c)+56*a^8*(-1/7*sin(d*x+c)^4*cos(d*x+c)^3-4/35*sin(d*x+c)^2*cos(d*x+c)^3-8/105*cos(d*x+c)^3)+70*a^8*(-1/
6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*cos(d*x+c)^3*sin(d*x+c)+1/16*cos(d*x+c)*sin(d*x+c)+1/16*d*x+1/16*c)+56*a^8*(-1
/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+28*a^8*(-1/4*cos(d*x+c)^3*sin(d*x+c)+1/8*cos(d*x+c)*sin(d*x+c)
+1/8*d*x+1/8*c)-8/3*a^8*cos(d*x+c)^3+a^8*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 0.987901, size = 431, normalized size = 1.65 \begin{align*} -\frac{1720320 \, a^{8} \cos \left (d x + c\right )^{3} - 16384 \,{\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 344064 \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 2408448 \,{\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 21 \,{\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 5880 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 235200 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 564480 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 161280 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{645120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/645120*(1720320*a^8*cos(d*x + c)^3 - 16384*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 1
05*cos(d*x + c)^3)*a^8 + 344064*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^8 - 2408448*(3*c
os(d*x + c)^5 - 5*cos(d*x + c)^3)*a^8 - 21*(96*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c -
45*sin(8*d*x + 8*c) - 120*sin(4*d*x + 4*c))*a^8 + 5880*(64*sin(2*d*x + 2*c)^3 - 120*d*x - 120*c + 3*sin(8*d*x
+ 8*c) + 24*sin(4*d*x + 4*c))*a^8 + 235200*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^8 -
564480*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^8 - 161280*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^8)/d

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Fricas [A]  time = 1.90641, size = 390, normalized size = 1.49 \begin{align*} \frac{71680 \, a^{8} \cos \left (d x + c\right )^{9} - 921600 \, a^{8} \cos \left (d x + c\right )^{7} + 3096576 \, a^{8} \cos \left (d x + c\right )^{5} - 3440640 \, a^{8} \cos \left (d x + c\right )^{3} + 765765 \, a^{8} d x + 63 \,{\left (128 \, a^{8} \cos \left (d x + c\right )^{9} - 4976 \, a^{8} \cos \left (d x + c\right )^{7} + 28328 \, a^{8} \cos \left (d x + c\right )^{5} - 46510 \, a^{8} \cos \left (d x + c\right )^{3} + 12155 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/80640*(71680*a^8*cos(d*x + c)^9 - 921600*a^8*cos(d*x + c)^7 + 3096576*a^8*cos(d*x + c)^5 - 3440640*a^8*cos(d
*x + c)^3 + 765765*a^8*d*x + 63*(128*a^8*cos(d*x + c)^9 - 4976*a^8*cos(d*x + c)^7 + 28328*a^8*cos(d*x + c)^5 -
46510*a^8*cos(d*x + c)^3 + 12155*a^8*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 40.576, size = 1018, normalized size = 3.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((7*a**8*x*sin(c + d*x)**10/256 + 35*a**8*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 35*a**8*x*sin(c + d
*x)**8/32 + 35*a**8*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 35*a**8*x*sin(c + d*x)**6*cos(c + d*x)**2/8 + 35*a
**8*x*sin(c + d*x)**6/8 + 35*a**8*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 105*a**8*x*sin(c + d*x)**4*cos(c + d
*x)**4/16 + 105*a**8*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 7*a**8*x*sin(c + d*x)**4/2 + 35*a**8*x*sin(c + d*x)
**2*cos(c + d*x)**8/256 + 35*a**8*x*sin(c + d*x)**2*cos(c + d*x)**6/8 + 105*a**8*x*sin(c + d*x)**2*cos(c + d*x
)**4/8 + 7*a**8*x*sin(c + d*x)**2*cos(c + d*x)**2 + a**8*x*sin(c + d*x)**2/2 + 7*a**8*x*cos(c + d*x)**10/256 +
35*a**8*x*cos(c + d*x)**8/32 + 35*a**8*x*cos(c + d*x)**6/8 + 7*a**8*x*cos(c + d*x)**4/2 + a**8*x*cos(c + d*x)
**2/2 + 7*a**8*sin(c + d*x)**9*cos(c + d*x)/(256*d) - 79*a**8*sin(c + d*x)**7*cos(c + d*x)**3/(384*d) + 35*a**
8*sin(c + d*x)**7*cos(c + d*x)/(32*d) - 8*a**8*sin(c + d*x)**6*cos(c + d*x)**3/(3*d) - 7*a**8*sin(c + d*x)**5*
cos(c + d*x)**5/(30*d) - 511*a**8*sin(c + d*x)**5*cos(c + d*x)**3/(96*d) + 35*a**8*sin(c + d*x)**5*cos(c + d*x
)/(8*d) - 16*a**8*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 56*a**8*sin(c + d*x)**4*cos(c + d*x)**3/(3*d) - 49*a
**8*sin(c + d*x)**3*cos(c + d*x)**7/(384*d) - 385*a**8*sin(c + d*x)**3*cos(c + d*x)**5/(96*d) - 35*a**8*sin(c
+ d*x)**3*cos(c + d*x)**3/(3*d) + 7*a**8*sin(c + d*x)**3*cos(c + d*x)/(2*d) - 64*a**8*sin(c + d*x)**2*cos(c +
d*x)**7/(35*d) - 224*a**8*sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - 56*a**8*sin(c + d*x)**2*cos(c + d*x)**3/(3*
d) - 7*a**8*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 35*a**8*sin(c + d*x)*cos(c + d*x)**7/(32*d) - 35*a**8*sin(c
+ d*x)*cos(c + d*x)**5/(8*d) - 7*a**8*sin(c + d*x)*cos(c + d*x)**3/(2*d) + a**8*sin(c + d*x)*cos(c + d*x)/(2*
d) - 128*a**8*cos(c + d*x)**9/(315*d) - 64*a**8*cos(c + d*x)**7/(15*d) - 112*a**8*cos(c + d*x)**5/(15*d) - 8*a
**8*cos(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)**8*cos(c)**2, True))

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Giac [A]  time = 1.24364, size = 235, normalized size = 0.9 \begin{align*} \frac{2431}{256} \, a^{8} x + \frac{a^{8} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} - \frac{33 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac{51 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac{17 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{8 \, d} - \frac{221 \, a^{8} \cos \left (d x + c\right )}{16 \, d} + \frac{a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac{59 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac{527 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{561 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{663 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

2431/256*a^8*x + 1/288*a^8*cos(9*d*x + 9*c)/d - 33/224*a^8*cos(7*d*x + 7*c)/d + 51/40*a^8*cos(5*d*x + 5*c)/d -
17/8*a^8*cos(3*d*x + 3*c)/d - 221/16*a^8*cos(d*x + c)/d + 1/5120*a^8*sin(10*d*x + 10*c)/d - 59/2048*a^8*sin(8
*d*x + 8*c)/d + 527/1024*a^8*sin(6*d*x + 6*c)/d - 561/256*a^8*sin(4*d*x + 4*c)/d - 663/512*a^8*sin(2*d*x + 2*c
)/d